-
8/2/2019 Use of Hft to Find Poles
1/4
Using the Half Fourier Transform for SEM analysis of both Early
and Late TimeResponses In the Presence of Noise
Tapan K Sarkar 1 , and Magdalena Salazar Palma 2
1Department of Electrical Engineering and Computer Science,
Syracuse University, Syracuse, New York, USA,[email protected]
2Departmento de Teoria de la Senal y Communicaciones,
Universidad Carlos III de Madrid, Leganes, Madrid,
[email protected]
Abstract
A technique for estimating the SEM parameters of damped
sinusoids utilizing both early and late time transientscattering
data contaminated by noise is described using the Half Fourier
Transform (HFT). The importance of thisnovel methodology is how to
simultaneously exploit both early time and late time data as for a
practical system it isdifficult to separate them and still be able
to identify the late time poles along with the early time specular
type of returns.
1. Introduction
The singularity expansion method (SEM) proposed by Baum has been
applied to quantify an electromagneticresponse in an expansion of
complex resonances of the system. It has been shown that the
dominant complex naturalresonances of a system are a minimal set of
parameters that define the overall physical properties of the
system. So, atransient scattering response is analyzed in terms of
the damped oscillations corresponding to the complex
resonantfrequency of the scatterer or target. In general, the
signal model of the observed late time of an
electromagnetic-energy-scattered response from an object can be
written as
,0);()exp()()()(1
T t t nt s Rt nt xt y m M
mm (1)
where )(t y = observed time domain response, )(t n = noise in
the data,)(t x = signal, m R = residues or complex amplitudes, mmm
j s ,
m = damping factors, m = angular frequencies ( mm f 2 ).After
sampling, the time variable, t is replaced by kT s, where T s is
the sampling period. The sequence can be
rewritten as M
m s
k mm s s s kT n z RkT nkT xkT y
1)()()()( for k=0,., N -1, (2)
smmT sm T je z si exp for m=1,2,, M . (3)Since the resonances
describe global wave fields that encompass the scattering object as
a whole, the SEM series
representation encounters convergence problem when applied to
the early time response of the objects. Early timeresponse is
strongly dependent on the nature of the source, the location of the
source, and the location of the observer.Usually the early time
response shows impulse-like characteristics. Because of this
difficulty, most previous techniquesused just late time signals
only. To excite the early time response a narrow Gaussian pulse is
selected. A Gaussian pulseis an entire function and is quite
adequate to describe pulse-like components in early time. Complex
exponentials areused to describe the late time signals. The concept
of a Turn-on time is utilized to consider a time when the
fullyexcited resonance can be used formally. So, the transient
scattered field can be modeled a
( )Scattered field Early Time Late Time Gaussian Pulses Damped
Sinusoids u t
where )(t u is a unit step function and is Turn-on time. But the
boundary in between early time and late time is notclear and
actually there is an intermediate zone in which the early time
pulse-like component and the late time dampedsinusoids are coupled
together. So the Turn-on time can be determined by an optimization
routine.
Since the resonance describe global wave fields that encompass
the scattering object as a whole, the SEM seriesrepresentation
encounters convergence difficulties at early times when portions of
the objects are not yet excited.Because of this difficulty, most
previous techniques used just late time signals only. The SEM
representation does notaccount for the impulsive portion of the
early time system responses behavior.
978-1-4244-5118-0/11/$26.00 2011 IEEE
-
8/2/2019 Use of Hft to Find Poles
2/4
Here, the Half Fourier Transform is used for parameter
optimization, because the parameter is continuous andthe
mathematical representations of the Half Fourier Transform (HFT) of
functions are clear and simple. The functionalrepresentation is
almost the same except for the coefficient and the sign in the
exponent. It means that their contributions in the Half Fourier
Transform domain are almost the same. That is not true in the
ordinary Fourier transform domain. Usually signals that we
encounter in real life are causal, that is, 0)(t x if t 0. And the
initialtime as to when a resonant component starts is different
from component to component. So it is possible to assume thateach
resonant component has a different Turn-on time.
2. Calculation of the HFT of an arbitrary waveform
The generalized Fourier Transform operator F by, 2 2/ 2 / 2( ) (
) x jn xn n F e H x e e H x where is avariable parameter, , with
negative values of corresponding to the inverse transform and )(t H
n is the
nth order Hermite polynomial. Therefore, the usual Fourier
Transform operator can be written as 2/ F to denote the
eigenvalue 2/ jne associated with the orthogonal Hermite
functions )(2/2
x H e n x . The associate Hermite (AH)
polynomials ),( t hn are defined in terms of the Hermite
polynomials ),( t H n through2
2
1( , ) exp , 022 !
n nn
t t h t H nn
; where is a scaling factor. The Hermite polynomials are
generated recursively through
2/
0
2
)(t e
t h ;
2
2)(
2/
1
2t tet h ; )(1)(21)( 21 t hnt th
nt h nnn ,
for 2n . The AH polynomials are orthonormal to each other and
form a complete set of basis in the interval
, . If )(t x is a piecewise smooth function defined on a finite
interval p p, and 2 2 ( )t e x t dt < ,
then )(t x can be expanded using the AH series as0
( ) ( ),n nn
x t a h t for t ; with
dt t ht xa nn )()( . Therefore, ( )( ) F x t u 0
( ). jnn na e h u The Half Fourier Transform (HFT) can be
obtained by substituting with 4/ .
3. Optimization
Complex natural frequencies occur in complex conjugate pairs and
they lie in the left half plane with a nonzeroreal part. To
represent real signals we treat two conjugate poles together. So,
the scattered field can be represented as
2
1 1
1( ) exp
2 2m m m mm
M N j t j t t n
m m n nm n
t B x t c e e e u t A C
where 0,0 t , 0m . mc s and m s are the amplitudes and the
phases, respectively. m s and m s are
the damping factors and the angular frequencies. n A s and n B s
are amplitudes and time shift of the Gaussian pulses.
nC s are coefficients which represent the pulse width. M is the
number of damped sinusoidal signals and N is thenumber of Gaussian
pulses.
To apply a parameter identification algorithm, the parameter set
is defined by
N N N M M M M M C A BC A Bcc p 11111111 ; while the residual
vector is defined by
-
8/2/2019 Use of Hft to Find Poles
3/4
2
2),(),(
21
t uGt uGr R
where )](;)([),( t xu X t uG and )](;)([),( t xu X t uG R R R .
)(u X is the HFT of the measured signal
)(t x , and ),( t uG R is the reconstructed half Fourier
transform. Both the original signal and its HFT are used to
compute the optimized parameters.2
2defines the squared 2 norms. )(u X is pre-calculated with the
time domain
scattered field. From the Half Fourier Transform of a shifted
Gaussian pulse and a damped sinusoid with a Turn-on
time, )(u X R is constructed as
uB B B
uC
C j Bu
C
C jC
j A
vu j
ee
vu j
ee j
cu X
n
n
n
n
n
N
nn
mm
v j
j
M
mmm
v j
jm R
mm
mm
22
22
21
12
exp2
21exp
1
122
exp
122
exp2
1
21
)(
2
2
21
22
22
11
21
2
22
21
where jvm1 , jvm2 , mmm vu j 11 2 , and mmm vu j 22 2 .
4. Example
The example presented is a wire scatterer. The time domain
transient electro-magnetic scattering responsesfrom various objects
have been calculated using an inverse Fourier transform of the
frequency domain data using anumerical electromagnetics code called
HOBBIES [2]. Noise is added to the time domain data. Generally
speaking,noise is more dominant in later time rather than at early
times. In this study, signal to noise ratio (SNR) is defined
as,
2 2
1 1
2 21010 log /
t t
i ii t i t
SNR s t n t
, where )(t s is a signal and )(t n is noise. Time domain
scattered field
is obtained using the same procedure with that of the previous
section and then white Gaussian noise is added with a
zero mean and with a finite SNR. The noise contaminated signal
and the Half Fourier Transform of the noisy signal isused to
identify the complex resonant frequency of the object. An average
error is taken from twenty trials for eachSNR.
The thin wire scatterer of length L and diameter d, which is
excited by an incident pulse of electromagneticradiation. As shown
in Figure 1, the length of the wire scatterer is 50 mm and the
aspect ratio (L/d) is 100. The incidentfield is coming from 45 from
the wire axis and is polarized with respect to the theta direction.
In this case 7 dampedsinusoids and 5 Gaussian pulses are used to
fit the time domain and the Half Fourier Transform domain data.
Noise isadded to the time domain data in the range between 10% and
40%. The order of the expansion for the associate Hermite
basis functions to carry out the Half Fourier Transform is
determined using the time-bandwidth product (2BT+1) rule,where B is
the bandwidth in the frequency domain and T is the time duration of
the signal. It means that to approximatea given waveform of
duration T and practically band limited by B (one-sided bandwidth),
using an orthonormal set of
basis functions in the time domain, at least (2BT+1) pieces of
basis are necessary from a mathematical point of view. Inthis
example the frequency band B is 100 GHz and the total time duration
T is 5 nsec. Therefore, to achieve this time-
bandwidth product for the backward scattered field one needs
approximately N = (2 100 5 + 1) = 1001 coefficients of
the Hermite expansion. Figure 2 represents the backward
scattered electromagnetic field without noise in the timedomain.
Figure 3(a) plots the time domain noisy signal with a SNR=10 dB and
Figure 3(b) is the reconstructed signalafter optimizations. Twenty
trials are performed for each SNR and then an average estimate of
the error is computed.
The root mean square error defined by the 21
( ) ( ) / N
o i r ii
RMS Error y t y t N where o y is the noisy time
domain data, r y is reconstructed data using optimized
parameters and N is the total number of data samples.
-
8/2/2019 Use of Hft to Find Poles
4/4
